Dynamical Systems and Bifurcation Theory

In urban communities, infrastructures that support living are indispensable. There is increased interest in alternative ways of providing such support systems, including semi-autonomous infrastructures resulting from the self-organization of local actors. In this study, we analyze the emergence and management of such infrastructures in light of the theory of complex adaptive systems, within which they are called ‘inverse infrastructures’. Empirical evidence is drawn from the case of water cooperatives in the town of Ikaalinen, Finland. Our analysis shows that, with favorable preconditions in place, inverse infrastructures may contribute significantly to local infrastructure services and so also to the functioning of society.

"The vision of Occidentalist International System into its historical evolution about the Ingurgitate Process of others living imperial social-orders and them intercultural systems of all over the world, since the historical seed of Welfen victory against the Waiblingen social order of Friedrich II's Holy Roman Empire (1250), up till the intetional delegitimization of last living Imperial Social Order of Japan after the second world war (1948).
Into last "great biforcation" of the second post-war process of anglo-saxonization around the world, the real centre of international anglo-saxon powership of London conduct the river water of spontaneous history inside mechanical channels of  binds inside the international system.
The univoque and convergent trend of decadence process of International System go toward the unique possible realization of an Imperial Social Order, in both possiblity as Dark Empire inside feudal elites dinasties, faked by democratic institutions, or as Light Empire of human family unity inside a new emergent and sincronical aristocratic hierarchy restauration of the ancient imperial social order, on the path of ethical common values inside civilizations, because other solutions can be only disasters. As though is been demonstrated by Raymond Aron, french author of International Relation Studies, the Empire is the only possible social order that really can assure a real and durable time of peace for the peoples, yesterday as today too. From this general vision, the containment of thesis develop his perspective from a New Confucianist Epistemology of science, as application of a defined complexity epistemology of dynamical systems and its bifurcation theory, analyzing the historical vectors of Occidentalism, Colonialism, Capitalism as the triad of a unique historical social process vortex started into a european medieval time that, contrary to the academic mainstream of actual storiography, is a process that isnt been never close but again living under different forms, as the medieval survivals istitutions of Europe as the studies of Ervin Laszlo on complexity of historical social orders can demonstrate about a substantial degeneration of social evolution respect the more complex social systems of traditional empires, in Europe as in all over the world."

This article presents a summary of applications of chaos and fractals in robotics. Firstly, basic concepts of determin‐ istic chaos and fractals are discussed. Then, fundamental tools of chaos theory used for identifying and quantifying chaotic dynamics will be shared. Principal applications of chaos and fractal structures in robotics research, such as chaotic mobile robots, chaotic behaviour exhibited by mobile robots interacting with the environment, chaotic optimization algorithms, chaotic dynamics in bipedal locomotion and fractal mechanisms in modular robots will be presented. A brief survey is reported and an analysis of the reviewed publications is also presented.

Die Entstehung der Komplexitätsforschung 3
Fragestellungen zu Fraktalen                 4
Cantor Staub                         5
Sierpinski Dreiecke – Teppiche         6
Schneeflocken – Kochsche Kurven         7
Phasenraum                         8
Seltsame Attraktoren                         9
Mandelbrot Menge – Apfelmännchen         10
Julia Mengen                         11
Bifurkationen                         12 – 13
Die Katastrophentheorie                 14
„Am Rand des Chaos“                 15
Die Glockenkurve                         16

17o Πανελλήνιο Συνέδριο Ένωσης Ελλήνων Φυσικών (Θεσσαλονίκη, 15-18 Μαρτίου 2018)
ΠΕΡΙΛΗΨΗ
Στην εργασία, που αποτελεί μέρος πτυχιακής υπό την επίβλεψη της επίκουρης καθηγήτριας Ευθυμίας Μελετλίδου, παρουσιάζεται η μελέτη του επιδημιολογικού μοντέλου που χρησιμοποιείται στην εργασία "A Core Group Model for disease transmission" των K. P. Hadeler και C. Castill-Chavez.  Συγκεκριμένα, παρουσιάζεται η διαδικασία μοντελοποίησης και ανάλυσης ενός κλειστού πληθυσμιακού συστήματος, η βιολογική του σημασία και τα δημογραφικά αποτελέσματα που προκύπτουν από την υποβολή του σε προφυλακτικά/προληπτικά ή θεραπευτικά/εκπαιδευτικά προγράμματα. Στην παραλλαγή του μοντέλου S.I.R που χρησιμοποιήθηκε, το δυναμικό σύστημα εμπλέκει τις πληθυσμιακές ομάδες με τον ρυθμό γεννήσεων, ανάρρωσης και θανάτων της εκάστοτε ομάδας, τον ρυθμό μετάδοσης της ασθένειας και τον ρυθμό εμβολιασμού των ασθενών. Η εργασία μελετά τον προσδιορισμό των συνθηκών ισορροπίας του συστήματος, την ευστάθειά τους και τις διακλαδώσεις του. Τέλος, τονίζεται η σημασία του φαινομένου των διακλαδώσεων αναφορικά με την επιλογή της προληπτικής ή θεραπευτικής μεθόδου που θα υιοθετηθεί ανάλογα με τις τιμές των παραμέτρων.

Conference presentation
17o Πανελλήνιο Συνέδριο Ένωσης Ελλήνων Φυσικών (Θεσσαλονίκη, 15-18 Μαρτίου 2018)
ΠΕΡΙΛΗΨΗ
Στην εργασία, που αποτελεί μέρος πτυχιακής υπό την επίβλεψη της επίκουρης καθηγήτριας Ευθυμίας Μελετλίδου, παρουσιάζεται η μελέτη του επιδημιολογικού μοντέλου που χρησιμοποιείται στην εργασία "A Core Group Model for disease transmission" των K. P. Hadeler και C. Castill-Chavez.  Συγκεκριμένα, παρουσιάζεται η διαδικασία μοντελοποίησης και ανάλυσης ενός κλειστού πληθυσμιακού συστήματος, η βιολογική του σημασία και τα δημογραφικά αποτελέσματα που προκύπτουν από την υποβολή του σε προφυλακτικά/προληπτικά ή θεραπευτικά/εκπαιδευτικά προγράμματα. Στην παραλλαγή του μοντέλου S.I.R που χρησιμοποιήθηκε, το δυναμικό σύστημα εμπλέκει τις πληθυσμιακές ομάδες με τον ρυθμό γεννήσεων, ανάρρωσης και θανάτων της εκάστοτε ομάδας, τον ρυθμό μετάδοσης της ασθένειας και τον ρυθμό εμβολιασμού των ασθενών. Η εργασία μελετά τον προσδιορισμό των συνθηκών ισορροπίας του συστήματος, την ευστάθειά τους και τις διακλαδώσεις του. Τέλος, τονίζεται η σημασία του φαινομένου των διακλαδώσεων αναφορικά με την επιλογή της προληπτικής ή θεραπευτικής μεθόδου που θα υιοθετηθεί ανάλογα με τις τιμές των παραμέτρων.

This paper presents a study of how different vibration modes contribute to the dynamics of an inclined cable that is parametrically excited close to a 2:1 internal resonance. The behaviour of inclined cables is important for design and analysis of cable-stayed bridges. In this work the cable vibrations are modelled by a four-mode model. This type of model has been used previously to study the onset of cable sway motion caused by internal resonances which occur due to the nonlinear modal coupling terms. A bifurcation study is carried out with numerical continuation techniques applied to the scaled and averaged modal equations. As part of this analysis, the amplitudes of the cable vibration response to support inputs is computed. These theoretical results are compared with experimental measurements taken from a 5.4 m long inclined cable with a vertical support input at the lower end. In general this comparison shows a very high level of agreement.

Several endomorphisms of a plane have been constructed by coupling two logistic maps. Here we study the dynamics occurring in one of them, a twisted version due to J. Dorband, which (like the other models) is rich in global bifurcations. By use of critical curves, absorbing and invariant areas are determined, inside which global bifurcations of the attracting sets (fixed points, closed invariant curves, cycles or chaotic attractors) take place. The basins of attraction of the absorbing areas are determined together with their bifurcations.

A dynamical system description of the transition process in shear flows with no linear instability starts with knowledge of exact coherent solutions, among them traveling waves (TWs) and relative periodic orbits (RPOs). We describe a numerical method to find such solutions in pipe flow and apply it in the vicinity of a Hopf bifurcation from a TW which looks to be especially relevant for transition. The dominant structural feature of the RPO solution is the presence of weakly modulated streaks. This RPO, like the TW from which it bifurcates, sits on the laminar-turbulent boundary separating initial conditions which lead to turbulence from those which immediately relaminarize.

1. Vorwort ………………….................……….......………..……. 1
2. einige Definitionen ……………................…….…………… 1
3. Strukturiertheit ………………………….................….……... 2
4. Crosskatalytische Dynamik ……………..........…….….. 2
5. Internet Links zu Peter Addors Publikationen ….. 4

One of the goals of nuclear power systems design and operation is to restrict the possible states of certain critical subsystems to remain inside a certain bounded set of admissible states and state variations. In the framework of an analytic or numerical modeling process of a BWR power plant, this could imply first to find a suitable approximation to the solution manifold of the system of nonlinear partial differential equations describing the stability behavior, and then a classification of the different solution types concerning their relation with the operational safety of the power plant. Inertial manifold theory gives a foundation for the construction and use of reduced order models (ROM's) of reactor dynamics to discover and characterize meaningful bifurcations that may pass unnoticed during digital simulations done with full scale computer codes of the nuclear power plant. The main aspects of approximate inertial manifolds and forms are briefly reviewed in the introduction of the paper.A complete numerical study of reactor dynamics using a realistic ROM currently involves the digital simulation of the behavior of approximately twenty state variables interrelated by a corresponding system of coupled nonlinear ordinary differential equations. The success of hybrid analytical-numerical bifurcation codes to detect interesting behavior, such as global bifurcations in BWR's, may be enhanced by studying suitable simplifications of ROM's, that is ROM's of ROM's. A previous generalization of the classical March-Leuba's model of BWR is briefly reviewed and a nonlinear integral-differential equation in the logarithmic power is derived. The asymptotic method developed by Krilov, Bogoliubov and Mitropolsky (KBM) is applied to obtain approximate equations of evolution for the amplitude and the phase of a manifold of oscillatory solutions jointly with a relation between an offset and the abovementioned amplitude. First, to exemplify the method working with a simpler problem, the KBM tentative solution (ansatz) is applied to construct approximate solutions of, and to study local bifurcations in, a van der Pol equation with continuous and discrete distribution of time delays. Then, the afore-mentioned ansatz is applied to the full nonlinear integral-differential equation of the BWR model. Analytical formulae are derived for the offset, the rate of change in the phase (the instantaneous frequency of oscillation) and the rate of change in the amplitude of oscillation, given as functions of the amplitude and the model parameters (steady state power and coolant flow, temperature and void reactivity coefficients, fuel to coolant heat transfer coefficient and other parameters from neutronics and thermal hydraulics). The obtained analytical formulae are applied to start a semi-analytical, mainly qualitative, approach to bifurcations and stability of the steady states located in different regions of parameters space. This includes a qualitative discussion of the possibility of both, super and subcritical Poincaré-Andronov-Hopf bifurcations, as well as a Bautin's bifurcation scenario. The preliminary qualitative results outlined in this study are consistent with results of recent digital simulations done with a full-scale reduced order model of BWR (PSI-TU Valencia-TU Dresden) and with the results obtained with the application of hybrid approaches to bifurcation theory done with the simplified March-Leuba's model of BWR.

Recent experimental evidence on the clustering of glutamate and GABA transporters on astrocytic processes surrounding synaptic terminals pose the question of the functional relevance of the astrocytes in the regulation of neural activity. In this perspective, we introduce a new computational model that embeds recent findings on neuron–astrocyte coupling at the mesoscopic scale intra- and inter-layer local neural circuits. The model consists of a mass model for the neural compartment and an astrocyte compartment which controls dynamics of extracellular glutamate and GABA concentrations. By arguments based on bifurcation theory, we use the model to study the impact of deficiency of astrocytic glutamate and GABA uptakes on neural activity. While deficient astrocytic GABA uptake naturally results in increased neuronal inhibition, which in turn results in a decreased neuronal firing, deficient glutamate uptake by astrocytes may either decrease or increase neuronal firing either transiently or permanently. Given the relevance of neuronal hyperexcitability (or lack thereof) in the brain pathophysiology, we provide biophysical conditions for the onset identifying different physiologically relevant regimes of operation for astrocytic uptake transporters.

We define a family B(t) of compact subsets of the unit interval which generalizes the sets of numbers whose continued fraction expansion has bounded digits. We study how the set B(t) changes as one moves the parameter t, and see that the family undergoes period-doubling bifurcations and displays the same transition pattern from periodic to chaotic behavior as the usual family of quadratic polynomials. The set E of bifurcation parameters is a fractal set of measure zero. We also show that the Hausdorff dimension of B(t) varies continuously with the parameter, and the dimension of each individual set equals the dimension of a corresponding section of the bifurcation set E.

This article presents a summary of applications of chaos and fractals in robotics. Firstly, basic concepts of deterministic chaos and fractals are discussed. Then, fundamental tools of chaos theory used for identifying and quantifying chaotic dynamics will be shared. Principal applications of chaos and fractal structures in robotics research, such as chaotic mobile robots, chaotic behaviour exhibited by mobile robots interacting with the environment, chaotic optimization algorithms, chaotic dynamics in bipedal locomotion and fractal mechanisms in modular robots will be presented. A brief survey is reported and an analysis of the reviewed publications is also presented.

This paper develops a dynamic model of North-South trade in which environment plays an important role. Our model is based on Chichilnisky North-South model for the macroeconomic interaction between two sectors of the world economy. The latter was introduced in a static context. We introduce dynamics in the original North-South model by allowing endogenous accumulation of capital. As a second extension of [1], we introduce here a variable which represents the system of property rights on the environmental asset which is used as an input to production. This could represent, for example, the property rights on forests from which wood is extracted to be used as an input to the production of traded goods or the property rights on water which is similarly used, perhaps for agricultural goods for export. The paper explains mathematically and through simulations the dynamics of a two-region world. There are two produced goods and two inputs to production. We show that as we vary the property rights of the environment the dynamics of the system changes. The less well defined are the property rights, the more chaotic are the model's dynamics.

A class of recurrent neural networks is investigated in the vicinity of the Bogdanov–Takens bifurcation point in the parameter space when the slope of the transfer function of the neurons at the origin is not equal to one. It will be shown that two different bifurcation diagrams can be constructed. In each bifurcation diagram, there are critical values for the parameters of the network for which curves of pitchfork and Hopf bifurcation intersect each other at a point where the linear part of the system that describes the network, has a pair of simple zero eigenvalues. As curves of homoclinic and heteroclinic bifurcation emanate from the Bogdanov–Takens point, a complicated behavior is observed by the variation of weights in the recurrent neural network.

The biological models - particularly the ecological ones - must be under- stood through the bifurcations they undergo as the parameters vary. However, the transition between two dynamical behaviours of a same system for diverse values of parameters may be sometimes quite involved. For instance, the analysis of the non generic motions near the transition states is the first step to understand fully the bifurcations occurring in complex dynamics.
In this article, we address the question to describe and explain a double bursting behaviour occurring for a tritrophic slow–fast system. We focus therefore on the appearance of a double homoclinic bifurcation of the fast subsystem as the predator death rate parameter evolves.
The first part of this article introduces the slow–fast system which extends Lotka–Volterra dynamics by adding a superpredator. The second part displays the analysis of singular points and bifurcations undergone by fast dynamics. The third part is devoted to the flow analysis near the homoclinic points. Finally, the fourth part is concerned with the main results about the existence of periodic orbits of different periods as the two homoclinic orbits are close enough to each other.

We extend and improve the existing characterization of the dynamics of general quadratic real polynomial maps with coefficients that depend on a single parameter λ and generalize this characterization to cubic real polynomial maps, in a consistent theory that is further generalized to real mth degree real polynomial maps. In essence, we give conditions for the stability of the fixed points of any real polynomial map with real fixed points. In order to do this, we have introduced the concept of canonical polynomial maps which are topologically conjugate to any polynomial map of the same degree with real fixed points. The stability of the fixed points of canonical polynomial maps has been found to depend solely on a special function termed Product Position Function for a given fixed point. The values of this product position determine the stability of the fixed point in question, when it bifurcates and even when chaos arises, as it passes through what we have termed stability bands. The exact boundary values of these stability bands are yet to be calculated for regions of type greater than one for polynomials of degree higher than three.